Your data matches 41 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000386
(load all 6 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000256
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 93%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [1] => [1,0]
 => []
 => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => []
 => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => []
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => []
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => []
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1]
 => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [5,4,2]
 => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1]
 => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0]
 => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => ? = 1
[1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1]
 => ? = 1
[1,1,1,1,1,0,0,1,0,0,0,0]
 => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [5,4,3]
 => ? = 1
[1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => ? = 0
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1]
 => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [5,4,2]
 => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1]
 => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [5,4,3]
 => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1]
 => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [5,4,3]
 => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2]
 => ? = 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1]
 => ? = 0
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [6,3,2]
 => ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,1,7] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1]
 => ? = 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [6,4,1]
 => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,1,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [6,4,2]
 => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [6,4,2,1]
 => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [6,4,1]
 => ? = 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [6,4,3]
 => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [6,4,3,1]
 => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,1,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
 => [6,4,3]
 => ? = 2
[1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,1,7] => [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2]
 => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,1]
 => ? = 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,1,2,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [6,3,2]
 => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,1,2,7] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [6,3,2,1]
 => ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,1,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [6,5,1]
 => ? = 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
 => [5,3,4,2,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [6,5,2]
 => ? = 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1]
 => ? = 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,1,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [6,5,1]
 => ? = 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [6,5,3]
 => ? = 2
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [6,5,3,1]
 => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,1,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,1,6,7] => [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
 => [6,5,3]
 => ? = 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,1,6,7] => [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2]
 => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,1]
 => ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [4,5,3,1,2,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [6,5,1]
 => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
 => [5,3,4,1,2,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [6,5,2]
 => ? = 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [6,5,2,1]
 => ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [6,5]
 => ? = 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => [4,5,2,1,3,6,7] => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [6,5,1]
 => ? = 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001083
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 89%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1,0]
 => [1] => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,2] => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [3,1,2] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [2,3,1] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [6,1,7,2,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,6,1,7,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,1,6,7,2,3,4] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,7,1,2,3] => ? = 0
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [6,1,7,2,3,4,5] => ? = 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [5,6,1,2,7,3,4] => ? = 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [5,1,6,2,7,3,4] => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [4,5,6,1,7,2,3] => ? = 1
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [4,5,6,2,3,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [5,6,1,2,7,3,4] => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [5,1,2,6,7,3,4] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [1,5,2,6,7,3,4] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [4,5,1,6,7,2,3] => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [5,1,2,6,7,3,4] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [4,1,5,6,7,2,3] => ? = 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [6,1,7,2,3,4,5] => ? = 1
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,6,1,7,2,3,4] => ? = 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,1,6,7,2,3,4] => ? = 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,1,2] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,7,1,2,3] => ? = 0
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [6,1,7,2,3,4,5] => ? = 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,1,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 2
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,1,7] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,5,6,1,2,7,3] => ? = 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,7,4] => ? = 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,1,7] => [1,1,1,1,0,0,1,0,0,1,0,0,1,0]
 => [1,5,2,6,3,7,4] => ? = 3
[1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,1,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
 => [4,5,1,6,2,7,3] => ? = 2
[1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,1,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,7,4] => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,1,7] => [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
 => [4,1,5,6,2,7,3] => ? = 2
[1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,6,1,7,2] => ? = 1
[1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,1,2,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [6,1,7,2,3,4,5] => ? = 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,1,2,7] => [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
 => [1,5,6,2,3,7,4] => ? = 2
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,1,2,7] => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,5,6,1,2,7,3] => ? = 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,1,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 2
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,1,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [4,5,1,2,6,7,3] => ? = 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,1,6,7] => [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
 => [4,1,5,2,6,7,3] => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,5,1,6,7,2] => ? = 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => [4,3,5,1,2,6,7] => [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
 => [1,5,2,3,6,7,4] => ? = 2
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => [3,4,5,1,2,6,7] => [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
 => [4,5,1,2,6,7,3] => ? = 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,1,2,5,6,7,3] => ? = 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [3,4,1,5,6,7,2] => ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => [4,1,2,5,6,7,3] => ? = 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,4,5,6,7,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [7,1,8,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [7,1,2,8,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [7,1,2,8,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,2,7,8,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,2,7,8,3,4,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,2,8,3,4,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [7,1,8,2,3,4,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,2,3,4,8,5,6,7] => ? = 1
Description
The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St000647
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 86%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1,0]
 => [1] => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,2] => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,1,3] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,2,3] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [3,1,2] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,3,1,2,4,6,7] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,3,2,1,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [6,3,2,1,4,5,7] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,2,7,1] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
 => [6,4,2,1,3,5,7] => ? = 2
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,2,7,1] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => [6,4,1,2,3,5,7] => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,7,1] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [6,4,3,1,2,5,7] => ? = 2
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,2,7,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [6,4,3,2,1,5,7] => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [5,4,6,2,3,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [6,3,1,2,4,5,7] => ? = 2
[1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [4,5,6,2,3,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [6,3,2,1,4,5,7] => ? = 1
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
 => [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [5,4,3,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2,1,3,4,7] => ? = 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [5,3,4,2,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,3,5,2,6,7,1] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [6,5,3,1,2,4,7] => ? = 2
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [6,5,3,2,1,4,7] => ? = 1
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [6,2,1,3,4,5,7] => ? = 1
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,5,2,3,6,7,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [6,5,2,1,3,4,7] => ? = 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [5,3,2,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [3,4,2,5,6,7,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => [6,5,4,2,1,3,7] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
 => [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [6,5,1,2,3,4,7] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [4,2,3,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [6,5,4,1,2,3,7] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,1,2,7] => ? = 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => ? = 1
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,1,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [4,3,1,2,5,6,7] => ? = 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,1,2] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,3,1,2,4,6,7] => ? = 2
[1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,1,2] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [5,3,2,1,4,6,7] => ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,2,1,3,4,6,7] => ? = 1
[1,1,0,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1
[1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,1,2] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [5,4,2,1,3,6,7] => ? = 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,1,2] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,1,2,3,6,7] => ? = 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,1,2] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [5,4,3,1,2,6,7] => ? = 1
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [4,2,1,3,5,6,7] => ? = 1
Description
The number of big descents of a permutation. For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$. The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].
Matching statistic: St001085
(load all 8 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 70%●distinct values known / distinct values provided: 60%
Values
[1,0]
 => [1] => [1,0]
 => [1] => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [2,1] => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,2] => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,3,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [5,4,6,3,7,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,7,2,1] => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => [4,5,6,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [5,6,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,2,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => [4,5,3,6,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [6,3,4,5,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [5,3,4,6,7,2,1] => [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
 => [5,4,3,6,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [4,3,5,6,7,2,1] => ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [6,7,5,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [6,5,7,4,2,3,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [5,6,7,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [6,7,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [6,5,4,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => [5,6,4,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => [6,4,5,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [5,4,6,7,2,3,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
 => [5,4,6,7,3,2,1] => ? = 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,2,3,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [4,5,6,7,3,2,1] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [6,7,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [6,5,7,3,2,4,1] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,6,7,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,4,3,2,1] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [6,7,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [6,4,5,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,4,6,3,2,7,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [5,4,6,3,2,7,1] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [4,5,6,3,2,7,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [4,5,6,3,2,7,1] => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [6,7,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [6,7,5,4,3,2,1] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [5,6,4,3,2,7,1] => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [6,4,3,5,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => [6,5,4,3,2,7,1] => ? = 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001036
(load all 8 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [7,8,4,5,6,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,2,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,2,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,2,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [7,8,4,3,5,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,2,3,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,2,3,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,2,3,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,2,3,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,2,3,8,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,2,4,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0
Description
The number of inner corners of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001037
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [7,8,4,5,6,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,2,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,2,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,2,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [7,8,4,3,5,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,2,3,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,2,3,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,2,3,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,2,3,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,2,3,8,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,2,4,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001732
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,8,2,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,2,3,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,2,3,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,2,3,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [7,8,4,5,6,2,3,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,2,3,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,2,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,3,2,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,2,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,2,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,2,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,2,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,3,2,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,2,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,2,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [7,8,4,3,5,2,6,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,2,3,4,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,2,3,4,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,2,3,4,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,2,3,4,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,2,3,5,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => [5,6,7,4,2,3,8,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,2,3,8,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,2,4,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 0 + 1
Description
The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.
Matching statistic: St000201
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => [.,.]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [[.,.],.]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [.,[.,.]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [[.,[.,.]],.]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [[[.,.],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [[.,.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [.,[.,[.,.]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[[.,.],[.,.]],.]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[.,[.,[.,.]]],.]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [[[.,[.,.]],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[.,[.,.]],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [[[[.,.],.],.],.]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [[[.,.],.],[.,.]]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[.,.],[.,[.,.]]]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [.,[.,[.,[.,.]]]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [[[[.,.],.],[.,.]],.]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [[[.,.],[.,[.,.]]],.]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [[[[.,.],[.,.]],.],.]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [[[[.,[.,.]],.],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[.,.],.],.],.],.]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [[[[.,.],.],.],[.,.]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[.,[.,.]]],[.,.]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [[[[[[[.,.],.],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [[[[[[[.,.],.],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [7,6,8,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[[[[[[.,.],.],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,5,6,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[[[[[[.,.],.],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,8,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [[[[[[.,.],[.,.]],[.,.]],.],.],.]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [7,6,4,5,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[[[[[[.,.],.],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,5,4,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[[[[[[.,.],.],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [7,4,5,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [[[[[[[.,.],.],.],[.,.]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [[[[[[[.,.],.],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [[[[[[.,[.,.]],[.,.]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [6,5,7,8,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,6,8,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,5,6,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,6,4,5,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,5,4,6,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
 => [7,4,5,6,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [7,6,8,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [[[[[[[.,.],[.,.]],.],.],.],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [7,6,5,3,4,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [7,6,4,3,5,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [7,5,4,3,6,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [7,8,3,4,5,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [[[[[[[.,[.,.]],.],.],.],.],.],.]
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [7,6,3,4,5,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,6,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [7,3,4,5,6,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [[[[[[[.,.],.],.],.],[.,.]],.],.]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [4,3,5,6,7,8,2,1] => [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
 => [[[[.,.],[.,[.,[.,[.,.]]]]],.],.]
 => ? = 1 + 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000052
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 80%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => 0
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [7,6,8,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,7,5,4,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [8,5,6,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,5,6,4,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [6,5,7,4,8,3,2,1] => [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [8,6,4,5,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [7,6,4,5,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,3,2,1] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [8,5,4,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [7,5,4,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [7,4,5,6,8,3,2,1] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [6,4,5,7,8,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
 => [6,5,7,8,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [8,6,7,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [7,6,8,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [8,7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [8,6,5,4,3,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,5,6,4,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [8,7,4,5,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,6,4,5,3,8,2,1] => [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [8,5,4,6,3,7,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
The following 31 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000069The number of maximal elements of a poset. St000007The number of saliances of the permutation. St000522The number of 1-protected nodes of a rooted tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000646The number of big ascents of a permutation. St000619The number of cyclic descents of a permutation. St000356The number of occurrences of the pattern 13-2. St000455The second largest eigenvalue of a graph if it is integral. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St000092The number of outer peaks of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000731The number of double exceedences of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St001960The number of descents of a permutation minus one if its first entry is not one. St000058The order of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000664The number of right ropes of a permutation. St001394The genus of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.